What is the quotient when x^3 - 5x^2 + 3x - 8 is divided by x - 3?

 To find the quotient when the polynomial \(x^3 - 5x^2 + 3x - 8\) is divided by \(x - 3\), we can use polynomial long division or synthetic division. Here, we'll use long division.

1. Write down the division, setting \(x^3 - 5x^2 + 3x - 8\) as the dividend inside the division symbol and \(x - 3\) as the divisor outside the division symbol.

2. Look at the first term of the dividend (\(x^3\)) and the first term of the divisor (\(x\)) and ask: "What do we multiply by \(x\) to get \(x^3\)?" The answer is \(x^2\), so write \(x^2\) above the division symbol, aligned with \(x^3\).

3. Multiply the divisor \(x - 3\) by \(x^2\) and write the result (\(x^3 - 3x^2\)) under the corresponding terms of the dividend.

4. Subtract this result from the dividend by changing the signs and adding: \(x^3 - 5x^2\) minus \(x^3 - 3x^2\) is \(-2x^2\). Bring down the next term of the dividend, which is \(+3x\), to get \(-2x^2 + 3x\).

5. Repeat the process: \(x\) multiplied by what gives \(-2x^2\)? The answer is \(-2x\). Write this above the division symbol next to \(x^2\), so you now have \(x^2 - 2x\).

6. Multiply \(x - 3\) by \(-2x\) to get \(-2x^2 + 6x\) and subtract this from \(-2x^2 + 3x\) to get \(-3x\). Bring down the next term, which is \(-8\), so now you have \(-3x - 8\).

7. Repeat again: \(x\) multiplied by what gives \(-3x\)? The answer is \(-3\). Write this above the division symbol next to \(-2x\), making your partial quotient \(x^2 - 2x - 3\).

8. Multiply \(x - 3\) by \(-3\) to get \(-3x + 9\), and subtract from our current remainder \(-3x - 8\) to find the new remainder. \(-3x - 8\) minus \(-3x + 9\) gives \(-17\).

Now, \(x^3 - 5x^2 + 3x - 8\) divided by \(x - 3\) gives a quotient of \(x^2 - 2x - 3\) with a remainder of \(-17\). Thus, the final answer can be expressed as \((x^2 - 2x - 3) + \frac{-17}{x - 3}\), but since you only asked for the quotient, the quotient is \(x^2 - 2x - 3\).